lecture 14 -- 3d update equations with pmlemlab.utep.edu/ee5390fdtd/lecture 14 -- 3d update...
TRANSCRIPT
10/20/2016
1
Lecture 14 Slide 1
EE 5303
Electromagnetic Analysis Using Finite‐Difference Time‐Domain
Lecture #14
3D Update Equations with Uniaxial PML These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited
Lecture Outline
Lecture 14 Slide 2
• Review of Lecture 13
• Conversion to the time‐domain
• Numerical approximations
• Derivation of the update equations
• Summary of all update equations
10/20/2016
2
Lecture 14 Slide 3
Review of Lecture 13
Lecture 14 Slide 4
No PAB in Two DimensionsIf we model a wave hitting some device or object, it will scatter the applied wave into potentially many directions. Waves at different angles travel at different speeds through a boundary. Therefore, the PAB condition can only be satisfied for one direction, not all directions.
fastest
slowest
x
y
10/20/2016
3
Lecture 14 Slide 5
Anisotropy to the Rescue!!It turns out we can prevent reflections at all angles, all frequencies, and for all polarizations if we allow our absorbing material to be diagonally anisotropic.
and
and
x
y
Lecture 14 Slide 6
The PML Parameters (1 of 2)We need to generalize our PML for waves incident on all boundaries.
0 0
0 0
0 0
y z
x
x z
y
x y
z
s s
s
s sS
s
s s
s
0
0
0
1
1
1
xx
yy
zz
xs x
j
ys y
j
zs z
j
10/20/2016
4
Lecture 14 Slide 7
The PML Parameters (2 of 2)The loss terms should increase gradually into the PMLs.
33 3
0 0 0 2 2 2x y z
x y z
x y zx y z
t L t L t L
? length of the PMLL
x x y y x
y
Lecture 14 Slide 8
Maxwell’s Equations with PML
1
0
0 0 0
1
0
0 0 0
0 0
1 1 1
1 1 1
1 1 1
y yzx zx
xx
y x zx zy
yy
yx
EEcj H
j j j y z
E Ecj H
j j j z x
jj j
1
0
0
y xzz
zz
E EcH
j x y
x xx x
y yy y
z zz z
D E
D E
D E
1
00 0 0 0
1
00 0 0 0
0 0
1 1 1
1 1 1
1 1
y yzx xxzx x
y yyx zx zy y
yx
HHj D c E
j j j y z
H Hj D c E
j j j z x
jj j
1
00 0
1 y xz zzz z
H HD c E
j x y
0
rE j S Hc
rD E
00
jH E S D
c
10/20/2016
5
Lecture 14 Slide 9
Conversion to the Time‐Domain
Lecture 14 Slide 10
Assume No Loss (=0)
1
0
0 0 0
1
0
0 0 0
0 0
1 1 1
1 1 1
1 1 1
y yzx zx
xx
y x zx zy
yy
yx
EEcj H
j j j y z
E Ecj H
j j j z x
jj j
1
0
0
y xzz
zz
E EcH
j x y
x xx x
y yy y
z zz z
D E
D E
D E
1
00 0 0
1
00 0 0
1
0 0 0
1 1 1
1 1 1
1 1 1
y yzx zx
y x zx zy
yx z
HHj D c
j j j y z
H Hj D c
j j j z x
jj j j
0
y xz
H HD c
x y
0
rE j s Hc
rD E
00
jH E s D
c
10/20/2016
6
Lecture 14 Slide 11
Recall Fourier Transform Properties
1
aa
a
t
g t G
ag t aG
dg t j G
dt
g d Gj
F
F
F
F
Lecture 14 Slide 12
Prepare Maxwell’s Equations for Conversion (1 of 2)
1
0
0 0 0
1 1 1y yzx zx
xx
EEcj H
j j j y z
We will start with this equation. The procedure for all other equations has the same basic steps.
The equation now becomes
First, let
yzEx
EEC
y z
1
0
0 0 0
1 1 1y Ex zx x
xx
cj H C
j j j
10/20/2016
7
Lecture 14 Slide 13
Prepare Maxwell’s Equations for Conversion (2 of 2)
The inverse term on the left is brought to the right.
The equation is then expanded by multiplying all of the terms.
1
0
0 0 0
0
0 0 0
1 1 1
1 1 1
y Ex zx x
xx
y Exzx x
xx
cj H C
j j j
cj H C
j j j
0 02
0 0 0
1 1y z y z E Exx x x x x
xx xx
c cj H H H C C
j j
For easy conversion to the time‐domain, we want all the terms to be multiplied by (j)a.
Lecture 14 Slide 14
Convert Each Term to Time‐Domain Individually
020
0
00
1 1y z Ex
x
Exx
xx
yx xx
x
z Hc
Cj
HcCj
jH
0 0
0
0
2 2
0
0
0 0
0
0
1
1
E Ex
tE Ex x
y z y zx x
x xxx x
ty z y z
xx
x
x
x
x
x
x
xx
H H dj
H tj
c cC
c cC C
Ht
C
t
H
dj
H t
10/20/2016
8
Lecture 14 Slide 15
The Time‐Domain Equation with PML
020
0
00
1 1y z Ex
x
Exx
xx
yx xx
x
z Hc
Cj
HcCj
jH
0
0
0
002
tExx
x
ty E
xxx
xzzx y
xx
cCH d
cH t
H t
tC t d
20 0
0 0
0
ty z y zx
x x
tE Exx x
xx xx
H tH t H d
t
c cC t C d
Lecture 14 Slide 16
Summary of All Time‐Domain Equations
0 02
0 0 0
002
0 0 0
t t
y z y z y yz zxx x x
xx xx
t
yx z x zx z x zy y y
yy yy
E t EE t Ec cH t H t H d dt y z y z
cE t E t E EcH t H t H dt z x z x
0 02
0 0 0
t
t t
x y x y y yx xzz z z
zz zz
d
E t EE t Ec cH t H t H d dt x y x y
002
0 0 0
002
0 0 0
t t
y z y z y yz zxx x x
t t
yx z x zx z x zy y y
z
H t HH t HcD t D t D d c dt y z y z
cH t H t H HD t D t D d c dt z x z x
D tt
002
0 0 0
t t
x y x y y yx xzz z
H t HH t HcD t D d c d
x y x y
x xx x
y yy y
z zz z
D t E t
D t E t
D t E t
10/20/2016
9
Lecture 14 Slide 17
Numerical Approximations
Lecture 14 Slide 18
We Will Handle Each Term Separately
0 02
0 0 0
t ty z y zx E Ex
x x x xxx xx
H t c cH t H d C t C d
t
The first governing equation in the time‐domain was
The terms listed separately are:
0
20
0
0
0
Term 1:
Term 2:
Term 3:
Term 4:
Term 5:
x
y zx
ty z
x
Ex
xx
tExx
xx
H t
t
H t
H d
cC t
cC d
10/20/2016
10
Lecture 14 Slide 19
Term 1
We previously approximated this term using finite‐differences as follows.
, , , ,
2 2
i j k i j k
x xt t t txH HH t
t t
Lecture 14 Slide 20
Term 2
The trick here is that we need to interpolate the H field at time t. We do this by averaging the values at time t+t/2 and t-t/2.
, , , , , , , ,
2 2
0 0 2
i j k i j k i j k i j kH HH Hx xy z t t t ty z
x
H HH t
10/20/2016
11
Lecture 14 Slide 21
Term 3 (1 of 3)
We approximate the integral in this equation with a summation.
2 20 0
2, ,
220
t ty z y z
x x
H H t ti j ky z
x TT t
H d H d
H t
We have a problem. The summation includes up to t+t/2, but the integral only goes up to t. We are accidentally integrating too far. This happened because the summation is integrating a term that exists at the half time steps.
Lecture 14 Slide 22
Term 3 (2 of 3)
We pull out the last term in the summation.
2, ,
2, ,
2 220 0
, ,
22
20 2
H H t tti j ky z y z
x x TT t
H Hi j ky
t tz
x TT t
i j k
x t t
tH
H d H t
H t
We force the extracted term to only integrate over half of a time step.
2
, ,
2 2
, , ,
20 0
,
2 2
2 2
i j k i j k
x xt t tH H t tt
i j ky z y zx x T
T
t
t
H d HH H t
t
H field interpolated at time tHalf time step.
10/20/2016
12
Lecture 14 Slide 23
Term 3 (3 of 3)
We factor out the t term.
The end‐time on this summation implies that we will update the summation beforeupdating the H field.
, , , ,
2, ,2 2
2 220 0 4
i j k i j kH H t tt x x i j kt t t ty z y zx x T
T t
H HtH d H
Lecture 14 Slide 24
Term 4
We approximated this term using finite‐differences in a previous lecture.
, ,0 0
, ,
i j kE Ex xi j k t
xx xx
c cC t C
, 1, , , , , 1 , ,
, ,
i j k i j k i j k i j k
i j k z z y yE t t t tx t
E E E EC
y z
Recall that the curl term is
10/20/2016
13
Lecture 14 Slide 25
Term 5
We approximate the integral in this equation with a summation.
0 0
0 0
, ,
, ,0
, ,00
, ,
, ,0
, ,00
t tE Ex xx x
xx xx
i j kH t i j kx Exi j k T
Txx
i j kH t i j kx Exi j k T
Txx
c cC d C d
cC t
c tC
There are no problems with this summation because it is integrating a term that exists at integer time steps.
The end‐time on this summation implies that we will update the summation before
updating the E field.
Lecture 14 Slide 26
Putting it All Together
Putting all the terms together, the equation is
0 02
0 0 0
t ty z y zx E Ex
x x x xxx xx
H t c cH t H d C t C d
t
, , , ,, , , , , , , , , , , ,2
, ,2 2 2 2 2 2
220 02 4
i j k i j ki j k i j k i j k i j k i j k i j kH H H H t tx x x x x x i j ky zt t t t t t t t t t t ty z
x TT t
H H H H H HtH
t
, ,
, , , ,00, , , ,
00
i j kH ti j k i j kxE Ex xi j k i j kt T
Txx xx
c tcC C
10/20/2016
14
Lecture 14 Slide 27
Summary of All Numerical Equations
, , , , , , , ,, , , , ,, , , ,2 , ,, , , , , , 02 2 02 2
, ,220 0 2 2
1
2 4
i j k i j k i j k i j k ti j k i j k i jH Hi j k i j k tH H Ht tx x y zt t i j kx x i j k i j k i j ky z xt t t t Et tx x x xi j kT tt t
T t xx
H H tH H c tcH H H C
t
,
, ,
, ,00
, , , , , , , ,, , , , , , , ,2, , , , , ,2 2 02 2
222 20 0
1
2 4
kt i j kE
xi j k TTxx
i j k i j k i j k i j k ti j k i j k i j k i j k H H tH H t ty y x zt ty y i j k i j k i j kx zt t t tt ty y yt t T
T ty
C
H H tH H cH H H
t
, ,
, , , ,0
, , , ,0
0
, , , , , , , ,, , , ,, , , ,
, , , , , ,2 2 2 22
0 0 2 2
1
2 4
i j kH ti j k i j kyE Ey yi j k i j kt T
Ty yy
i j k i j k i j k i j ki j k i j k H Hi j k i j k H H t tz z x yt tz z i j k i j k i j kx yt t t tt tz z z Tt t
T
c tC C
H H tH HH H H
t
, ,2 , , , ,00
, , , ,2 00
t i j kt H ti j k i j kzE Ez zi j k i j kt T
t Tzz zz
c tcC C
, , , ,, , , , , , , ,, , , , , , 2
, , , , , , , , , ,0
0200 0 0 2
,
1
2 4
ti j k i j k ti j k i j k i j k i j ki j k i j k i j kD DD D Dty z i j k i j k i j k i j k i j kx x x xy z xH Ht t t t t tx x x x xt Tt t t T
TT t
i
y t t
tD D D D c tD D D c C C
t
D
, , , ,, , , , , , ,, , , , , , 2
, , , , , , , , , ,0
0200 0 0 2
, ,
1
2 4
ti j k i j k tj k i j k i j k i j ki j k i j k i j kD DD D Dtx z i j k i j k i j k i j k i j ky y yx z yH Ht t t ty y y y yt Tt t t T
TT t
i j k i
z zt t t
tD D D c tD D D c C C
t
D D
, , , ,, , , , , ,, , , , , , 2
, , , , , , , , , ,0
0200 0 0 2
1
2 4
ti j k i j k tj k i j k i j ki j k i j k i j kD DD D Dtx y i j k i j k i j k i j k i j kz zx y zH Ht t tz z z z zt Tt t t T
TT t
tD D c tD D D c C C
t
, , , ,, ,
, , , ,, ,
, , , ,, ,
i j k i j ki j k
x xx xt t t t
i j k i j ki j k
y yy yt t t t
i j k i j ki j k
z zz zt t t t
D E
D E
D E
Lecture 14 Slide 28
Derivation of the Update Equations
10/20/2016
15
Lecture 14 Slide 29
The Starting Point
Just as before, we solve the numerical equation for the future time value of H. This term appears three times and these must be collected.
, , , ,, , , , , , , ,
2 2 2 2
0
, , , ,
2, , , , , ,
2 2 220
, ,0 0, ,
0
2
1
4
i j k i j ki j k i j k i j k i j kH Hx x x xy zt t t t t t t t
i j k i j k tH H ty z i j k i j k i j k
x x xt t t t TT t
i j k xExi j k t
xx
H H H H
t
tH H H
c c tC
, ,
, ,
, ,
0
i j k tHi j kE
xi j k txx T
C
Lecture 14 Slide 30
Expand the Equation
We start by expanding the equation by multiplying all of the terms.
, , , , , , , ,
, , , ,
20 0 2
, , , , , , , , , ,
, , , ,
22
, ,
2 2
, ,
20 0 2
1 1
2 2
4 4
i j
i j k i j k i j k i j kH H H Hi j k i j ky z y z
tx xt t t
i j k i j k i j k i j k i j kH
k i j k
tx xt
H H H H Hy z y z y zi
t t
i j k
txt
j k
txt
H Ht t
t t
H
H
H
H
, ,
2, ,
220
, ,
, , , ,0 0, , , ,
0 0
i j k tt
i j k
x TT t
i j k tHi j k i j kxE E
x xi j k i j kt txx xx T
tH
c c tC C
, , , , , ,, , , ,, ,2
, , , ,2 22
20
, ,, ,
, ,2 2
20 2
1
2 4
i j k i j k i j k ti j k i j k H Hi j k tH H tx y ztx i j k i j k
i j ki j k tx
tx i j kt t y zt ttxt x Tt
tt
Tx
HHH
H
H Ht
H t
, ,
, , , ,0 0, , , ,
0 0
i j k tHi j k i j kxE E
x xi j k i j kt txx xx T
c c tC C
10/20/2016
16
Lecture 14 Slide 31
Collect Common Terms
Next we collect the coefficients of the common field terms.
, , , , , , , ,
, , , ,
20 0 2
, , , , , , , , , ,
, , , ,
22
, ,
2 2
, ,
20 0 2
1 1
2 2
4 4
i j
i j k i j k i j k i j kH H H Hi j k i j ky z y z
tx xt t t
i j k i j k i j k i j k i j kH
k i j k
tx xt
H H H H Hy z y z y zi
t t
i j k
txt
j k
txt
H Ht t
t t
H
H
H
H
, ,
2, ,
220
, ,
, , , ,0 0, , , ,
0 0
i j k tt
i j k
x TT t
i j k tHi j k i j kxE E
x xi j k i j kt txx xx T
tH
c c tC C
, , , ,, , , ,
20 0
, , , ,, , , ,
, ,
20 0 2
, , , ,
,
2
20
,1
2 4
1
2 4
i j k i j ki j k i j k H HH Hy zy z
i j k i j ki j k i j k H HH Hy z i j ky z
txt
i j k i j kH H
i j k
tx
y z
t
x
Ht
t
tH
t
tH
, ,2 , , , ,, , 0 0
, , , ,2 0 0
t i j kt tHi j k i j ki j k xE E
x xi j k i j kT t tT t xx xx T
c c tC C
Lecture 14 Slide 32
Solve for the Future H Field
Solving our numerical equation for the H field at t+t/2 yields
, , , ,, , , ,
2 , ,0 0
, , 0
, , , ,, , , , , , , ,2
20
2
0
,
0
,
12 4
1 1
2 4 2
i j k i j ki j k i j k H HH Hy zy z
i j ki j k xx
txi j k i j k ti j k i j k i j k i j kH HH
i j k
H H Hy z yy z y
txt
z
t
t cH
t
t
H
t
, ,
, , , ,
20
, ,
, , , ,0, , 2
, ,0 0, , , ,, , , ,
0
20 0
4
+
1 12 4
i j kExi j k i j k tH H
z
i j kHx i j k i j kH H
ti j k y zi j kxx E
xi j k i j k ti j k i j k H HH H HTy zy z y
Ct
c t t
Ct
t t
2
, ,
, , , ,, , , , 2
20 02 4
tt
i j k
xi j k i j k Ti j k i j k H HH T ty zz
Ht
This is the largest equation in the entire course!
10/20/2016
17
Lecture 14 Slide 33
Final Form of the Update Equation for Hx
, , , , , , , ,, , , , , , , ,
, , , ,
0 1 , ,2 20 0 0 00
, , 02 , , , ,
0
1 1 1
2 4 2 4
1
i j k i j k i j k i j ki j k i j k i j k i j kH H H HH H H Hy z y zi j k i j ky z y z
Hx Hx i j k
Hx
i j k
Hx i j k i j k
Hx xx
t tm m
t tm
cm
m
, ,
, , , ,, , , ,03 4, , , , , , 2
0 00 0
1 1
i j kHi j k i j ki j k i j kx H H
Hx Hx y zi j k i j k i j k
Hx xx Hx
c t tm m
m m
, ,, , , , , , , , , , , , , ,
1 2 3 42
, ,
2
i j ki j k i j k i j k i j k i j k i j k i j kEHx x Hx x H
i j k
x x CEx Hx Hxt t t ttt tm H m C m I m IH
, 1, , , , , 1 , ,2, , , ,, , , , , ,
220
t i j k i j k i j k i j ktti j k i j k z z y yi j k i j k i j kE E t t t t
tCEx x Hx x xt Tt ttT t
T
E E E EI C I H C
y z
The update coefficients are computed before the main FDTD loop.
The integration terms are computed inside the main FDTD loop, but before the update equation.
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
Lecture 14 Slide 34
Are These Equations Correct?
A simple way to test these update equations is to set the PML parameters to zero and see if the update equations reduce to what we discussed in a previous lecture.
The update coefficients reduce to
, , , , , , , ,01 2 3 4, ,1 0 0i j k i j k i j k i j k
Hx Hx Hx Hxi j k
xx
c tm m m m
, 1, , , , , 1 , ,
, , 0,
,2 ,2
,
i j k i j k i j k i j k
z z y yi j k t t t tx i
i j k
j k
xt
x
x t ttH
E E E Ec tH
y z
The update equation reduces to
For problems uniform in the z direction, the z derivative is zero. This equation reduces to exactly what is shown in Lecture 11.
, 1 ,
, 0,2
,
2
i j i j
z zi j t tx i jt t
j
x t txx
i E Ec tH
yH
10/20/2016
18
Lecture 14 Slide 35
Summary of the Update Equations
Lecture 14 Slide 36
Final Form of the Update Equation for Hx
, , , , , , , ,, , , , , , , ,
, , , ,
0 1 , ,2 20 0 0 00
, , 02 , , , ,
0
1 1 1
2 4 2 4
1
i j k i j k i j k i j ki j k i j k i j k i j kH H H HH H H Hy z y zi j k i j ky z y z
Hx Hx i j k
Hx
i j k
Hx i j k i j k
Hx xx
t tm m
t tm
cm
m
, ,
, , , ,, , , ,03 4, , , , , , 2
0 00 0
1 1
i j kHi j k i j ki j k i j kx H H
Hx Hx y zi j k i j k i j k
Hx xx Hx
c t tm m
m m
, ,, , , , , , , , , , , , , ,
1 2 3 422
, ,
2
i j ki j k i j k i j k i j k i j k i j k i j kEtHx x Hx x Hx CE
i j k
x Hx Hxt t tt tx t t
m H m C m I IH m
, 1, , , , , 1 , ,2, , , ,, , , , , ,
220
t i j k i j k i j k i j ktti j k i j k z z y yi j k i j k i j kE E t t t t
tCEx x Hx x xt Tt ttT t
T
E E E EI C I H C
y z
The update coefficients are computed before the main FDTD loop.
The integration terms are computed inside the main FDTD loop, but before the update equation.
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
10/20/2016
19
Lecture 14 Slide 37
Final Form of the Update Equation for Hy
, , , , , , , ,, , , , , , , ,
, , , ,
0 1 , ,2 20 0 0 00
, ,0
2 , , , ,
0
1 1 1
2 4 2 4
1
i j k i j k i j k i j ki j k i j k i j k i j kH H H HH H H Hx z x zi j k i j kx z x z
Hy Hy i j k
Hy
i j k
Hy i j k i j k
Hy yy
t tm m
t tm
cm
m
, ,
, , , ,, , , ,0
3 4, , , , , , 20 00 0
1 1
i j kHi j k i j ki j k i j ky H H
Hy Hy x zi j k i j k i j k
Hy yy Hy
c t tm m
m m
, ,, , , , , , , , , , , , , ,
1 2 3 4
,
2 2
,
2
+i j ki j k i j k i j k i j k i j k i j k i j kE
t tHy y Hy y Hy CEy Hy Hy
i j
t
k
t tt
ty tm H m C m I m IH
, , 1 , , 1, , , ,2, , , ,, , , , , ,
220
t i j k i j k i j k i j ktti j k i j ki j k i j k i j k x x z zE E t t t t
tCEy y Hy y yt t Tt tT t
T
E E E EI C I H C
z x
The update coefficients are computed before the main FDTD loop.
The integration terms are computed inside the main FDTD loop, but before the update equation.
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
Lecture 14 Slide 38
Final Form of the Update Equation for Hz
, , , , , , , ,, , , , , , , ,
, , , ,
0 12 , , 20 00 00
, , 02 , , , ,
0
1 1 1
2 24 4
1
i j k i j k i j k i j ki j k i j k i j k i j kH H H HH H H Hx y x yx y x yi j k i j k
Hz Hz i j k
Hz
i j k
Hz i j k i j k
Hz zz
t tm m
t tm
cm
m
, ,
, , , ,, , , ,03 4, , , , , , 2
0 00 0
1 1
i j kHi j k i j kzi j k i j k H H
Hz Hz x yi j k i j k i j k
Hz zz Hz
c t tm m
m m
, ,, , , , , , , , , , , , , ,
1 2 3 4
,
2 2
,
2
+i j ki j k i j k i j k i j k i j k i j k i j kE
Hz z t Hz z Hz CEz Hz Hz
i j k
z tt
tttt tm H m C m I m IH
1, , , , , 1, , ,2, , , ,, , , , , ,
220
t i j k i j k i j k i j ktti j k i j k y y x xi j k i j k i j kE E t t t t
tCEz z Hz z zt Tt ttT t
T
E E E EI C I H C
x y
The update coefficients are computed before the main FDTD loop.
The integration terms are computed inside the main FDTD loop, but before the update equation.
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
10/20/2016
20
Lecture 14 Slide 39
Final Form of the Update Equation for Dx
, , , , , , , ,, , , , , , , ,
, , , ,
0 1 , ,2 20 0 0 00
, , , ,02 3, ,
0
1 1 1
2 4 2 4
i j k i j k i j k i j ki j k i j k i j k i j kD D D DD D D Dy z y zi j k i j ky z y z
Dx Dx i j k
Dx
i j k i j
Dx Dxi j k
Dx
t tm m
t tm
cm m
m
, ,
, , , ,, ,0
4, , , , 20 00 0
1 1
i j kDi j k i j kk i j kx D D
Dx y zi j k i j k
Dx Dx
c t tm
m m
, , , ,, , , , , , , , , , , ,
1 2 3 42
, ,
2
+i j k i j ki j k i j k i j k i j k i j k i j kH
t tDx x Dx x Dx CHx D
i j
x Dx t tt
k
x t t t tm D m C m I m ID
2 2 2 2
2
, , , , 1, , , 1,2, ,, , , ,, , , ,
02 2
t t t t
t
t i j k i j kt i j k i j kt y yi j ki j k i j k z zi j k i j k t t t tH H
tCHx x Dx x xtT tTtT
T t
H HH HI C I D C
y z
The update coefficients are computed before the main FDTD loop.
The integration terms are computed inside the main FDTD loop, but before the update equation.
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
Lecture 14 Slide 40
Final Form of the Update Equation for Dy
, , , , , , , ,, , , , , , , ,
, , , ,
0 1 , ,2 20 0 0 00
, , , ,0
2 3, ,
0
1 1 1
2 4 2 4
i j k i j k i j k i j ki j k i j k i j k i j kD D D DD D D Dx z x zi j k i j kx z x z
Dy Dy i j k
Dy
i j k i j
Dy Dyi j k
Dy
t tm m
t tm
cm m
m
, ,
, , , ,, ,0
4, , , , 20 00 0
1 1
i j kDi j k i j kk i j ky D D
Dy x zi j k i j k
Dy Dy
c t tm
m m
, , , ,, , , , , , , , , , , ,
1 2 3 422
, ,+
i j k i j ki j k i j k i j k i j k i j k i j kHttDy y Dy y Dy CHy D
i j
y Dyt t tt
k
y t t tm D m C m I m ID
2 2 2 2
2
, , , , 1 , , 1, ,2, ,, , , ,, , , ,
022
t t t t
t
tt i j k i j k i j k i j k
t i j ki j k i j k x x z zi j k i j k t t t tH HtCHy y Dy y yt tT tT
TT t
H H H HI C I D C
z x
The update coefficients are computed before the main FDTD loop.
The integration terms are computed inside the main FDTD loop, but before the update equation.
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
10/20/2016
21
Lecture 14 Slide 41
Final Form of the Update Equation for Dz
, , , , , , , ,, , , , , , , ,
, , , ,
0 1 , ,2 20 0 0 00
, , , ,02 3, ,
0
1 1 1
2 4 2 4
i j k i j k i j k i j ki j k i j k i j k i j kD D D DD D D Dx y x yi j k i j kx y x y
Dz Dz i j k
Dz
i j k i j
Dz Dzi j k
Dz
t tm m
t tm
cm m
m
, ,
, , , ,, ,0
4, , , , 20 00 0
1 1
i j kDi j k i j kk i j kz D D
Dz x yi j k i j k
Dz Dz
c t tm
m m
, , , ,, , , , , , , , , , , ,
1 2 3 42
, ,
2
+i j k i j ki j k i j k i j k i j k i j k i j kH
t tDz z Dz z Dz CHz D
i j
z Dz t tt
k
z t t t tm D m C m I m ID
2 2 2 2
2
, , 1, , , , , 1,2, ,, , , ,, , , ,
02 2
t t t t
t
t i j k i j kt i j k i j kt y yi j ki j k i j k x xi j k i j k t t t tH H
tCHz z Dz z ztT tTtT
T t
H H H HI C I D C
x y
The update coefficients are computed before the main FDTD loop.
The integration terms are computed inside the main FDTD loop, but before the update equation.
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
Lecture 14 Slide 42
Final Update Equations for Ex, Ey, and Ez
, ,, , , ,
1 1 1, , , , , ,
1 1 1
i j ki j k i j k
Ex Ey Ezi j k i j k i j k
xx zzyy
m m m
, ,, ,
1
, ,, ,
1
, ,, ,
1
, ,
, ,
, ,
i j ki j k
Ex x t t
i j ki j k
i j k
x t
Ey y t t
i j ki j k
E
t
i j k
y t t
i j k
z z t tz t t
E
E
E
m D
m D
m D
The update coefficients are computed before the main FDTD loop.
The update equations are computed inside the main FDTD loop.